The third problem (all parts) will be graded by hand as usual. For the most part they do not involve much maple.
If you are on the list of people without maple just submit a text file as usual with all your work shown. Your papers will be graded by hand as usual.
If you work with other people on this homework, you must make note of the problems on which you collaborated, and who your collaborators were. Note that you are not allowed to turn a copy of someone else's work; you must type all your own work.
Text File: HW3.txt
Problem 1
> read("/afs/eos.ncsu.edu/users/k/kaltofen/www/courses/LinAlgebra/Maple/refpkg/initpkg.mpl"):
> with(refpkg):
> with(linalg):
You are given the following linear system:
2 x1 - x2 + x3 = -1
4 x1 + x2 + 2 x3 = 3
2 x1 + x2 + 2 x3 = -2
a) Find matrix A, and a vector b such that the system can be written as A x = b. (x = vector([x1,x2,x3]))
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> A := ; # put your answer as a matrix here.
> b := ; # put your answer as a vector here.
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b) Find matrices T and U, so that U is in REF and T A = U. (Hint: T will be the product of elementary matrices).
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> T := ; # put your answer here.
> U := ; # put your answer here.
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c) Find the L which gives a LU factorization of A (with U from part b).
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> L := ; # put your answer here.
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d) Using backsubstitution twice, solve the linear system, that is first solve Ly = b, then solve U x = y. Notice that this 'x' should solve LU x = Ax = b.
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> y := ; # put the result of the first back substitution here.
> x := ; # put your solution as a vector here. Note: evalm(A&*x) should be 'b' from part (a).
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Problem 2
> read("/afs/eos.ncsu.edu/users/k/kaltofen/www/courses/LinAlgebra/Maple/refpkg/initpkg.mpl"):
> with(refpkg):
> with(linalg):
You are given the following linear system:
2 x1 - x2 + x3 = -1
4 x1 + x2 + 2 x3 = 3
2 x1 + x2 + 2 x3 = -2
a) Find matrix A, and a vector b such that the system can be written as A x = b. (x = vector([x1,x2,x3])). Verify that A is non-singular by computing its determinant.
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> A := ; # put your answer as a matrix here.
> b := ; # put your answer as a vector here.
> detA := ; # put the determinant here.
b) We are going to use Cramer's rule to compute one entry of the inverse of A. If we want the entry in the third column of the second row (the [2,3] entry) we just need to compute the quotient of two determinants. First find the two matrices whose determinants will be the numerator and the denominator, then compute them to find the value of the entry.
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> numerator_matrix_b := ; # Write the matrix whose determinant is the numerator of the entry
> denominator_matrix_b := ; # Write the matrix whose determinant is the denominator of the entry
> entry23 := ; # put the value of the entry here
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c) Now we use Cramer's rule to find the solution to the linear system. If we want the second component of the solution (the x2 value) we will again be finding the quotient of two determinants.
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> numerator_matrix_c := ; # Write the matrix whose determinant is the numerator of the x2 value
> denominator_matrix_c := ; # Write the matrix whose determinant is the denominator of the x2 value
> x2_value := ; # put the value of the x2 component of the solution here
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Problem 3
You do not need to use Maple to do any of the following problems. In fact you may want to write the solutions out by hand, then type them into Maple. Though, Maple may be useful for checking your work.
Part 1
Prove that even when A B != B A that det (A B) = det (B A). You will have to appeal to a theorem here. Showing an example is not enough.
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Part 2
Prove that the following is true, or find a counter example to show it is not:
det (A + B) = det A + det B.
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Part 3
Let H be the set of points in the unit disc in the xy-plane. That is H = { (x,y) | x^2 + y^2 <= 1}. Let addition and scalar multiplication be defined as usual: (a,b) + (c,d) = (a+c,b+d), and alpha (a,b) = (alpha a, alpha b). Decide whether H is a vector space under these operations. (Note: to show it is, you must verify all the vector space axioms, while to show it is not, you need only show one of them does not hold.)
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Part 4
Consider the set of real ordered pairs (R^2) with the following operations:
(i) (u_1, u_2) + (v_1, v_2) = (u_1 * v_1, u_2 * v_2)
(ii) alpha * (u_1, u_2) = (alpha + u_1, alpha + u_2).
Demonstrate with specific examples that this set does not satisfy at least three vector space axioms.
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